Why do this problem?

A hook, in the form of images, manipulatives, or interactivity, draws students in to a task as their natural curiosity compels them to explore.

In this problem, a simple picture offers an intriguing challenge at several different levels, and the accompanying interactivity is engaging and enticing.

The problem is particularly valuable as it gives students an opportunity to work on a proof to explain why something is impossible (e.g. the 2-sandwich). 

As there are many solutions in the case of 7-sandwiches and 8-sandwiches, the problem provides an opportunity for many students to discover their own solution, different to any that have already been found.

Possible approach

It is helpful to have digits to rearrange. You may wish to print off two sets of Digit Cards for each group of students. Alternatively, if computers are available, students could use the interactivities.

Begin by introducing the problem with digits 1 to 3:

 

"In this arrangement there is one number sandwiched between the '1' cards, two numbers sandwiched between the '2' cards, but only one number sandwich between the '3' cards."
"Is it possible to make a complete sandwich with one number between the '1' cards, two numbers between the '2' cards, and three numbers between the '3' cards?"

Give students plenty of time to explore. When they find a solution, challenge them to try with 1-4, and then 1-7, and 1-8, recording carefully any solutions they find.

Once students have worked on the problem, invite a few students to write up their solutions on the board as a focus for discussion in a plenary. Focus on the following key questions:

Can you make 2-sandwiches? How do you know?
Are any sandwiches the same looked at in different ways?
How many different 3-sandwiches are there? What about 4-sandwiches?
Did anyone manage to find a 7-sandwich? Did anyone find a different one?
What about an 8-sandwich?

Possible extension

More Number Sandwiches invites students to explore the impossibility of constructing certain sandwiches, and leads to an elegant proof of why 5- and 6-sandwiches are impossible.

Possible support

The 3- and 4-sandwiches should be accessible for most students, especially if digit cards or the interactivity are used.